Seminarski zadatak iz Kvantne fizike
|
|
- August Parrish
- 5 years ago
- Views:
Transcription
1 Seminarski zadatak iz Kvantne fizike Vinko Šuria. velače 00. Fizički odsek Prirodoslovno - matematičkog fakulteta Sveučilišta u Zagrebu, Bienička, Zagreb, Hrvatska Zadatak 7. Neka e potencialna energia V (x, y, z) homogena funkcia koordinata stupna homogenosti ν, t. V (λx, λy, λz) = λ ν V (x, y, z) Dokažite da e sredna vriednost kinetičke energie u stanu diskretnog spektra vezana sa srednom vriednošću potencialne energie relaciom T = ν V
2 Sažetak Izvodi se virialni teorem u klasično mehanici, te hipervirialni i virialni teorem u kvantno fizici. Pokazue se negova valanost na primeru harmoničkog oscilatora i vodikovog atoma. Uvod U fizici često e vrlo teško i komplicirano računati potencialne i kinetičke energie nekih sustava, pa se traže relacie koe povezuu te velične kako bi olakšale nihovo tražene, er e onda potrebno samo izračunati ednu. Jedna od tih relacia e i virialni teorem koi u klasično mehanici povezue vremensko usrednene kinetičke i potencialne energie, a u kvantno fizici očekivanu vriednost kinetičke i potencialne energie. Korisnost toga e posebno izražena u kvantno fizici er e često puno lakše računati očekivanu vriednost potencialne energie od kinetičke. Prvo ću izvesti virialni teorem u klasično mehanici, a potom i u kvantno kako bi pokazao da su istog oblika. Za izvod u kvanto fizici prvo ću izvesti hipervirialni teorem koi tvrdi da vremenska promena vremenski nezavisnih operatora u stacionarnom stanu e ednaka nuli. Na krau ću pokazati valanost virialnog teorema na energetskim stanima kvantnog harmoničkog oscilatora i vodikova atoma. Virialni teorem u klasično mehanici U izvodima se korisiti Eulerov teorem o homogenim funkciama reda homogenosti ν koi glasi: f x = νf () x Krećem od kinetičke energie i pokazuem da e ona proporcionalna potencialno T = m x () No, kinetička energia e homogena funkcia u brzini reda, pa možemo pisati T = ( ) x p = d x p p x () dt
3 Primenimo. Newtonov zakon i pretpostavku da e sila konzervativna pa silu možemo napisati kao gradient potenciala i dobiemo ( ) T = d x p + V x () dt x U slučau da e potencial homogena funkcia reda ν pišemo ( ) T = d x p + νv (5) dt Vremenskim usrednenem u slučau omedenih ili periodičnih putana središni član isčezava te dolazimo do izraza za virialni teorem u klasično mehanici T = ν V (6) Ova teorem e valan u slučau da e sila konzervativna i u slučau omedenih ili periodičkih putana, što nisu vrlo strogi zahtevi pa ova teorem ima široki spektar primene. U klasično mehanici e račun kinetičke i potencialne energie relativno ednostavan, pa se ova teorem ne primenue često. Virialni teorem u kvantno fizici Činenica da se nalazimo u stanu diskretnog spektra nam govori da imamo kvantizaciu energetskih stana, što e i prirodno u kvantno fizici. Nadale biti će promatrana stacionarna stana, odnosno stana koa ne ovise o vremenu. To ograničene e blago er podrazumeva da e Hamiltonian sustava neovisan o vremenu, što e čest sluča. U dalnem tekstu podrazumievati ću da e Hamiltonian oblika H = p m + V. Hipervirialni teorem Neka e Hamiltonian sustava dan s H te negova stacionarna stana s Ψ koa zadovolavau vremenski ovisnu Schrödingerovu ednadžbu: HΨ = i t Ψ (7) Vremenska promiena očekivane vriednosti operatora A = Ψ A Ψ dana s d dt A = i A [H, A] + (8) t
4 Uz pretpostavku da operator A ne ovisi eksplicitno o vremenu kao ni Hamiltionian sustava, pa svostvne energie stana možemo označiti s E n i sa ψ n svostvena stana sustava. Za stacionarno stane ukupnu valnu funkciu pišemo: Ψ n = ψ n e i Ent (9) U stacionarnom stanu očekivana vriednost vremenski neovisnog operatora A e Ψ n A Ψ n = ψ n A ψ n (0) Jednadžba (8) u stacionarnom stanu postae d dt A = i ψ n [H, A] ψ n () = i ψ n HA AH ψ n () = i (E n E n ) ψ n A ψ n = 0 () i vidimo da očekivana vriednost vremenski neovisnog operatora A se u stacionarnom stanu ne miena u vremenu. Hipervirialni teorem tvrdi da e promena očekivane vriednosti vremenski neovisnog operatora u stacionarnom stanu edanka nuli. Virialni teorem U klasično mehanici vremenska derivacia veličine r p isčezava za periodičke ili omedene putane. Analogno u kvantno mehanici vremenska derivacia očekivane vriednosti operatora r p isčezava u stacionarnom stanu, što e ranie dokazano hipervirialnim teoremom, er niedan operator ne ovisi eksplicitno o vremenu. d (r p) r p = [r p, H] + = 0 () dt i t
5 [r p, H] = i i x i [p i, H] + [x i, H] p i (5) = ( x i i V ) ( i i + i x i m p i + i ) m p i p i (6) V i = x i + p i (7) x i m p i V i = x i (8) m x i Primenimo Eulerov teorem T = ν V (9) Time e virialni teorem dokazan. On vriedi u stacionarnom stanu u slučau da operator r p ne ovisi eksplicitno o vremenu i da e potencial homogena funkcia u kordinatama reda homogenosti ν. U slučau da potencial nie homogena funkciia, izraz (8) se svodi na Rasprava T = i x i V i x i (0) Pokazati ću valnost virialnog teorema za sluča D harmoničkog oscilatora i elektrona u vodikovom atomu za prvih 5 stana. U harmoničkom oscilatoru potencial e homogena funkcia reda ν =, dok u vodikovom atomu Columbov potencial e reda ν =. Takoder pokazuem i zbro kinetičke i potencialne energie radi usporedbe. Iz tablice se vidi da e očekivana vriednost kinetičke energie ednaka potencialno za harmonički oscilator, dok za vodikov atom su suprotnog predznaka i po apsolutnom iznosu razlikuu za faktor, kao što predvida virialni teorem. Kao primer korištena pokazati ću kako se ednostavno može izračunati očekivana vriednost kinetičke i potencialne energie za vodikov atom u stanu n = : T = V V = T () T + V = E T = E () V = E () Ovde vidimo kako se pomoću ednostavnog računa mogu izračunati očekivane vriednosti kinetičke i potencialne energie pomoću virialnog teorema poznavaući samo energiu nekog stana. Inače bi energie računali preko izraza T = ψ n T ψ n (analogno i za potencialnu energiu), što dae složene integrale koe često nie ednostavno riešiti.
6 Harmonički oscilator Vodikov atom n T n V n E n 0 ω ω ω 9 ω 9 ω 9 ω ω ω ω ω ω ω ω ω ω n T n V n E n I h I h I h 5 I h I h I h 9 I h 9 I h 9 I h 6 I h 8 I h 6 I h 5 I h 5 I h 5 I h Tablica : Usporedbe potencialne i kinetičke energie u harmoničkom oscilatoru i vodikovom atomu Zaklučak Pokazano e da virialni teorem u klasično mehanici i u kvantno fizici imau ednak oblik, samo se razlikuu u tome što klasični ima energie usrednene u vremenu dok kvantni očekivane vriednosti. Takoder e pokazan i hipervirialni teorem za koeg e bilo i intuitivno očekivati da vriedi. Na krau e pokazana i primena virialnog teorema, te se pomoču nega vrlo lako mogu izračunati očekivane vriednosti kinetičke i potencialne energie ukoliko znamo samo ukupnu energiu tog stacionarnog stana. Literatura [] H. Goldstein, C. Poole, J. Safko, Classical Mechanics third edition, Addison Wesley, 00 [] Şakir Erkoç, Fundamentals of quantum mechanics, Taylor & Francis Group, 007 [] David J. Griffiths, Introduction to Quantum Mechanics, Preatince Hall, 995 5
TEORIJA SKUPOVA Zadaci
TEORIJA SKUPOVA Zadai LOGIKA 1 I. godina 1. Zapišite simbolima: ( x nije element skupa S (b) d je član skupa S () F je podskup slupa S (d) Skup S sadrži skup R 2. Neka je S { x;2x 6} = = i neka je b =
More informationZANIMLJIV NAČIN IZRAČUNAVANJA NEKIH GRANIČNIH VRIJEDNOSTI FUNKCIJA. Šefket Arslanagić, Sarajevo, BiH
MAT-KOL (Banja Luka) XXIII ()(7), -7 http://wwwimviblorg/dmbl/dmblhtm DOI: 75/МК7A ISSN 5-6969 (o) ISSN 986-588 (o) ZANIMLJIV NAČIN IZRAČUNAVANJA NEKIH GRANIČNIH VRIJEDNOSTI FUNKCIJA Šefket Arslanagić,
More informationProjektovanje paralelnih algoritama II
Projektovanje paralelnih algoritama II Primeri paralelnih algoritama, I deo Paralelni algoritmi za množenje matrica 1 Algoritmi za množenje matrica Ovde su data tri paralelna algoritma: Direktan algoritam
More informationINTRODUCTION TO LOW FREQUENCY LOCAL PLASMONS IN BULK EXTRINSIC SEMICONDUCTORS UDC 538.9; Yuri Kornyushin
FACTA UNIVERSITATIS Series: Physics, Chemistry and Technology Vol. 2, N o 5, 2003, pp. 253-258 INTRODUCTION TO LOW FREQUENCY LOCAL PLASMONS IN BULK EXTRINSIC SEMICONDUCTORS UDC 538.9; 621.315.5 Yuri Kornyushin
More information2. RAZVOJ KVANTNE MEHANIKE (QM)
FRAKTALNA MEHANIKA Prof.dr Đuro Koruga LEKCIJA ODNOS KLASIČNE I KVANTNE MEHANIKE Da bi se razumeo odnos klasične (KM) i kvantne mehanike (QM) neophodno e poznavati nihov nastanak i razvo. Kratak pregled
More informationMUSICAL COMPOSITION AND ELEMENTARY EXCITATIONS OF THE ENVIRONMENT
Interdisciplinary Description of Complex Systems (-2), 22-28, 2003 MUSICAL COMPOSITION AND ELEMENTARY EXCITATIONS OF THE ENVIRONMENT Mirna Grgec-Pajić, Josip Stepanić 2 and Damir Pajić 3, * c/o Institute
More informationKLASIFIKACIJA NAIVNI BAJES. NIKOLA MILIKIĆ URL:
KLASIFIKACIJA NAIVNI BAJES NIKOLA MILIKIĆ EMAIL: nikola.milikic@fon.bg.ac.rs URL: http://nikola.milikic.info ŠTA JE KLASIFIKACIJA? Zadatak određivanja klase kojoj neka instanca pripada instanca je opisana
More informationpretraživanje teksta Knuth-Morris-Pratt algoritam
pretraživanje teksta Knuth-Morris-Pratt algoritam Jelena Držaić Oblikovanje i analiza algoritama Mentor: Prof.dr.sc Saša Singer 18. siječnja 2016. 18. siječnja 2016. 1 / 48 Sadržaj 1 Uvod 2 Pretraživanje
More informationSimulacija fluida tehnikom SPH
SVEUČILIŠTE U ZAGREBU FAKULTET ELEKTROTEHNIKE I RAČUNARSTVA ZAVRŠNI RAD br. 5494 Simulacia fluida tehnikom SPH Juri Kos Zagreb, lipan, 2018 Sadrža 1. Uvod... 2 2. Osnove mehanike fluida... 5 2.1 Svostva
More informationMetode praćenja planova
Metode praćenja planova Klasična metoda praćenja Suvremene metode praćenja gantogram mrežni dijagram Metoda vrednovanja funkcionalnosti sustava Gantogram VREMENSKO TRAJANJE AKTIVNOSTI A K T I V N O S T
More informationRed veze za benzen. Slika 1.
Red veze za benzen Benzen C 6 H 6 je aromatično ciklično jedinjenje. Njegove dve rezonantne forme (ili Kekuléove structure), prema teoriji valentne veze (VB) prikazuju se uobičajeno kao na slici 1 a),
More informationFIZIKALNA KOZMOLOGIJA VII. VRLO RANI SVEMIR & INFLACIJA
FIZIKALNA KOZMOLOGIJA VII. VRLO RANI SVEMIR & INFLACIJA KOZMIČKI SAT ranog svemira Ekstra zračenje u mjerenju CMB Usporedba s rezultatima LEP-a Usporedba CMB i neutrina Vj.: Pozadinsko zračenje neutrina
More informationQuantum Statistical Aspects of Charge Transfer on Electrodes~'
CR 0 AT IC A CB: EM l CA ACT A 44 (1972) 15 CCA-679 541.138 :530.145 Conference Paper Quantum Statistical Aspects of Charge Transfer on Electrodes' E. Bergmann BatteHe Institute, Advanced Studies Center,
More informationHamilton Jacobijeva formulacija klasične mehanike
Sveučilište J. J. Strossmayera u Osijeku Odjel za matematiku Vedran Šimošić Hamilton Jacobijeva formulacija klasične mehanike Diplomski rad Osijek, 2010. Sveučilište J. J. Strossmayera u Osijeku Odjel
More informationŠime Šuljić. Funkcije. Zadavanje funkcije i područje definicije. š2004š 1
Šime Šuljić Funkcije Zadavanje funkcije i područje definicije š2004š 1 Iz povijesti Dvojica Francuza, Pierre de Fermat i Rene Descartes, posebno su zadužila matematiku unijevši ideju koordinatne metode
More informationPhysics 106a, Caltech 13 November, Lecture 13: Action, Hamilton-Jacobi Theory. Action-Angle Variables
Physics 06a, Caltech 3 November, 08 Lecture 3: Action, Hamilton-Jacobi Theory Starred sections are advanced topics for interest and future reference. The unstarred material will not be tested on the final
More informationMetode izračunavanja determinanti matrica n-tog reda
Osječki matematički list 10(2010), 31 42 31 STUDENTSKA RUBRIKA Metode izračunavanja determinanti matrica n-tog reda Damira Keček Sažetak U članku su opisane metode izračunavanja determinanti matrica n-tog
More informationMANY ELECTRON ATOMS Chapter 15
MANY ELECTRON ATOMS Chapter 15 Electron-Electron Repulsions (15.5-15.9) The hydrogen atom Schrödinger equation is exactly solvable yielding the wavefunctions and orbitals of chemistry. Howev er, the Schrödinger
More informationAutomorphic Inversion and Circular Quartics in Isotropic Plane
Original scientific paper Accepted 0. 11. 008. EMA JURKIN Automorphic Inversion and Circular Quartics in Isotropic Plane Automorphic Inversion and Circular Quartics in Isotropic Plane ABSTRACT In this
More informationAlgoritam za množenje ulančanih matrica. Alen Kosanović Prirodoslovno-matematički fakultet Matematički odsjek
Algoritam za množenje ulančanih matrica Alen Kosanović Prirodoslovno-matematički fakultet Matematički odsjek O problemu (1) Neka je A 1, A 2,, A n niz ulančanih matrica duljine n N, gdje su dimenzije matrice
More informationON THE TWO BODY PROBLEM UDC (045)=20. Veljko A. Vujičić
FACTA UNIVERSITATIS Series: Mechanics, Automatic Control and Robotics Vol. 4, N o 7, 005, pp. 03-07 ON THE TWO BODY PROBLEM UDC 53.5(045)0 Veljko A. Vujičić Mathematical Institute, JANN, 00 Belgrade, p.p.
More informationAn Algorithm for Computation of Bond Contributions of the Wiener Index
CROATICA CHEMICA ACTA CCACAA68 (1) 99-103 (1995) ISSN 0011-1643 CCA-2215 Original Scientific Paper An Algorithm for Computation of Bond Contributions of the Wiener Index Istvan Lukouits Central Research
More informationPart III. The Synchro-Betatron Hamiltonian
Part III Kevin Li Synchro-Betatron Motion 43/ 71 Outline 5 Kevin Li Synchro-Betatron Motion 44/ 71 Canonical transformation to kinetic energy (1) Hamiltonian H(u, P u, s 0, H s ; s) = ( 1 + x ) (E 0 β0
More informationFormule za udaljenost točke do pravca u ravnini, u smislu lp - udaljenosti math.e Vol 28.
1 math.e Hrvatski matematički elektronički časopis Formule za udaljenost točke do pravca u ravnini, u smislu lp - udaljenosti Banachovi prostori Funkcija udaljenosti obrada podataka optimizacija Aleksandra
More informationGRAVITY. David J. Jeffery. Department of Physics, University of Idaho, PO Box , Moscow, Idaho , U.S.A January 1 ABSTRACT
GRAVITY David J. Jeffery Department of Physics, University of Idaho, PO Box 440903, Moscow, Idaho 83844-0903, U.S.A. 2008 January 1 ABSTRACT Lecture notes on what the title says. Subject headings: keywords
More informationSimple Harmonic Oscillator
Classical harmonic oscillator Linear force acting on a particle (Hooke s law): F =!kx From Newton s law: F = ma = m d x dt =!kx " d x dt + # x = 0, # = k / m Position and momentum solutions oscillate in
More informationUse precise language and domain-specific vocabulary to inform about or explain the topic. CCSS.ELA-LITERACY.WHST D
Lesson seven What is a chemical reaction? Science Constructing Explanations, Engaging in Argument and Obtaining, Evaluating, and Communicating Information ENGLISH LANGUAGE ARTS Reading Informational Text,
More informationThe Periodic Table. Periodic Properties. Can you explain this graph? Valence Electrons. Valence Electrons. Paramagnetism
Periodic Properties Atomic & Ionic Radius Energy Electron Affinity We want to understand the variations in these properties in terms of electron configurations. The Periodic Table Elements in a column
More informationOn the relation between Zenkevich and Wiener indices of alkanes
J.Serb.Chem.Soc. 69(4)265 271(2004) UDC 547.21:54 12+539.6 JSCS 3152 Original scientific paper On the relation between Zenkevich and Wiener indices of alkanes IVAN GUTMAN a*, BORIS FURTULA a, BILJANA ARSI]
More informationAb initio proučavanje neadijabatskih efekata kod malih molekula
Ab initio proučavanje neadijabatskih efekata kod malih molekula Marko Mitić Fakultet za fizičku hemiju, Univerzitet u Beogradu Seminar iz fizike/astrofizike Departman za fiziku, PMF, Novi Sad 15. april
More informationVARIATIONAL PRINCIPLE AND THE HYDROGEN ION: TWO PARAMETERS
VARIATIONAL PRINCIPLE AND THE HYDROGEN ION: TWO PARAMETERS Link to: physicspages home page. To leave a comment or report an error, please use the auxiliary blog. References: Griffiths, David J. 005), Introduction
More informationKrivulja središta i krivulja fokusa u pramenu konika. konika zadanom pomoću dviju dvostrukih točaka u izotropnoj ravnini
Stručni rad Prihvaćeno 18.02.2002. MILJENKO LAPAINE Krivulja središta i krivulja fokusa u pramenu konika zadanom pomoću dviju dvostrukih točaka u izotropnoj ravnini Krivulja središta i krivulja fokusa
More informationScripture quotations marked cev are from the Contemporary English Version, Copyright 1991, 1992, 1995 by American Bible Society. Used by permission.
N Ra: E K B Da a a B a a, a-a- a aa, a a. T, a a. 2009 Ba P, I. ISBN 978-1-60260-296-0. N a a a a a, a,. C a a a Ba P, a 500 a a aa a. W, : F K B Da, Ba P, I. U. S a a a a K Ja V B. S a a a a N K Ja V.
More informationRešenja zadataka za vežbu na relacionoj algebri i relacionom računu
Rešenja zadataka za vežbu na relacionoj algebri i relacionom računu 1. Izdvojiti ime i prezime studenata koji su rođeni u Beogradu. (DOSIJE WHERE MESTO_RODJENJA='Beograd')[IME, PREZIME] where mesto_rodjenja='beograd'
More informationCompact operators, the essential spectrum and the essential numerical range
Mathematical Communications (998), 0-08 0 Compact operators, the essential spectrum and the essential numerical range Damir Bakić Abstract. Some properties of bounded operators on Hilbert space concerned
More information5 questions, 3 points each, 15 points total possible. 26 Fe Cu Ni Co Pd Ag Ru 101.
Physical Chemistry II Lab CHEM 4644 spring 2017 final exam KEY 5 questions, 3 points each, 15 points total possible h = 6.626 10-34 J s c = 3.00 10 8 m/s 1 GHz = 10 9 s -1. B= h 8π 2 I ν= 1 2 π k μ 6 P
More informationODREĐIVANJE DINAMIČKOG ODZIVA MEHANIČKOG SUSTAVA METODOM RUNGE-KUTTA
Sveučilište u Zagrebu GraĎevinski faklultet Kolegij: Primjenjena matematika ODREĐIVANJE DINAMIČKOG ODZIVA MEHANIČKOG SUSTAVA METODOM RUNGE-KUTTA Seminarski rad Student: Marija Nikolić Mentor: prof.dr.sc.
More informationSpeed of light c = m/s. x n e a x d x = 1. 2 n+1 a n π a. He Li Ne Na Ar K Ni 58.
Physical Chemistry II Test Name: KEY CHEM 464 Spring 18 Chapters 7-11 Average = 1. / 16 6 questions worth a total of 16 points Planck's constant h = 6.63 1-34 J s Speed of light c = 3. 1 8 m/s ħ = h π
More informationA new Lagrangian of the simple harmonic oscillator
A new Lagrangian of the simple harmonic oscillator Faisal Amin Yassein Abdelmohssin 1 Sudan Institute for Natural Sciences, P.O.BOX 3045, Khartoum, Sudan Abstract A new Lagrangian functional of the simple
More informationSolid State Physics FREE ELECTRON MODEL. Lecture 17. A.H. Harker. Physics and Astronomy UCL
Solid State Physics FREE ELECTRON MODEL Lecture 17 A.H. Harker Physics and Astronomy UCL Magnetic Effects 6.7 Plasma Oscillations The picture of a free electron gas and a positive charge background offers
More informationCLASSIFICATION OF CONIC SECTIONS IN P E 2 (R) Jelena Beban-Brkić and Marija Šimić Horvath
RAD HAZU. MATEMATIČKE ZNANOSTI Vol. 18 = 519 (2014): 125-143 CLASSIFICATION OF CONIC SECTIONS IN P E 2 (R) Jelena Beban-Brkić and Marija Šimić Horvath Abstract. This paper gives a complete classification
More informationModified Zagreb M 2 Index Comparison with the Randi} Connectivity Index for Benzenoid Systems
CROATICA CHEMICA ACTA CCACAA 7 (2) 83 87 (2003) ISSN-00-3 CCA-2870 Note Modified Zagreb M 2 Index Comparison with the Randi} Connectivity Index for Benzenoid Systems Damir Vuki~evi} a, * and Nenad Trinajsti}
More informationANALYSIS OF THE RELIABILITY OF THE "ALTERNATOR- ALTERNATOR BELT" SYSTEM
I. Mavrin, D. Kovacevic, B. Makovic: Analysis of the Reliability of the "Alternator- Alternator Belt" System IVAN MAVRIN, D.Sc. DRAZEN KOVACEVIC, B.Eng. BRANKO MAKOVIC, B.Eng. Fakultet prometnih znanosti,
More informationUse precise language and domain-specific vocabulary to inform about or explain the topic. CCSS.ELA-LITERACY.WHST D
Lesson eight What are characteristics of chemical reactions? Science Constructing Explanations, Engaging in Argument and Obtaining, Evaluating, and Communicating Information ENGLISH LANGUAGE ARTS Reading
More informationNAPREDNI FIZIČKI PRAKTIKUM 1 studij Matematika i fizika; smjer nastavnički MJERENJE MALIH OTPORA
NAPREDNI FIZIČKI PRAKTIKUM 1 studij Matematika i fizika; smjer nastavnički MJERENJE MALIH OTPORA studij Matematika i fizika; smjer nastavnički NFP 1 1 ZADACI 1. Mjerenjem geometrijskih dimenzija i otpora
More informationTermodinamika. FIZIKA PSS-GRAD 29. studenog Copyright 2015 John Wiley & Sons, Inc. All rights reserved.
Termodinamika FIZIKA PSS-GRAD 29. studenog 2017. 15.1 Thermodynamic Systems and Their Surroundings Thermodynamics is the branch of physics that is built upon the fundamental laws that heat and work obey.
More informationFOURIEROVE PREOBRAZBE- PRIMJENA U FIZICI
SVEUČILIŠTE JOSIPA JURJA STROSSMAYERA U OSIJEKU ODJEL ZA FIZIKU Preddiplomski sveučilišni studij fizike FOURIEROVE PREOBRAZBE- PRIMJENA U FIZICI Završni rad Anton Aladenić Osijek, 2014. SVEUČILIŠTE JOSIPA
More informationUNIFORM PLASMA OSCILLATIONS IN ELLIPSOID OF CONDUCTIVE MATERIAL UDC Yuri Kornyushin
FACTA UNIVERSITATIS Series: Physics, Chemistry and Technology Vol. 3, N o 1, 2004, pp. 35-39 UNIFORM PLASMA OSCILLATIONS IN ELLIPSOID OF CONDUCTIVE MATERIAL UDC 533.92 Yuri Kornyushin Maître Jean Brunschvig
More informationThe Einstein A and B Coefficients
The Einstein A and B Coefficients Austen Groener Department of Physics - Drexel University, Philadelphia, Pennsylvania 19104, USA Quantum Mechanics III December 10, 010 Abstract In this paper, the Einstein
More informationZANIMLJIVI ALGEBARSKI ZADACI SA BROJEM 2013 (Interesting algebraic problems with number 2013)
MAT-KOL (Banja Luka) ISSN 0354-6969 (p), ISSN 1986-5228 (o) Vol. XIX (3)(2013), 35-44 ZANIMLJIVI ALGEBARSKI ZADACI SA BROJEM 2013 (Interesting algebraic problems with number 2013) Nenad O. Vesi 1 Du²an
More informationA SPECTRAL ATLAS OF λ BOOTIS STARS
Serb. Astron. J. 188 (2014), 75-84 UDC 524.3 355.3 DOI: 10.2298/SAJ1488075P Professional paper A SPECTRAL ATLAS OF λ BOOTIS STARS E. Paunzen 1 and U. Heiter 2 1 Department of Theoretical Physics and Astrophysics,
More informationA L A BA M A L A W R E V IE W
A L A BA M A L A W R E V IE W Volume 52 Fall 2000 Number 1 B E F O R E D I S A B I L I T Y C I V I L R I G HT S : C I V I L W A R P E N S I O N S A N D TH E P O L I T I C S O F D I S A B I L I T Y I N
More informationRESISTANCE PREDICTION OF SEMIPLANING TRANSOM STERN HULLS
Nenad, VARDA, University of Zagreb, Faculty of Mechanical Engineering and Naval Architecture, I. Lučića 5, 10000 Zagreb Nastia, DEGIULI, University of Zagreb, Faculty of Mechanical Engineering and Naval
More informationSTATISTICAL ANALYSIS OF WET AND DRY SPELLS IN CROATIA BY THE BINARY DARMA (1,1) MODEL
Hrvatski meteoroloπki Ëasopis Croatian Meteorological Journal, 4, 2006., 43 5. UDK: 55.577.22 Stručni rad STATISTICAL ANALYSIS OF WET AND DRY SPELLS IN CROATIA BY THE BINARY DARMA (,) MODEL Statistička
More informationto the potential V to get V + V 0 0Ψ. Let Ψ ( x,t ) =ψ x dx 2
Physics 0 Homework # Spring 017 Due Wednesday, 4/1/17 1. Griffith s 1.8 We start with by adding V 0 to the potential V to get V + V 0. The Schrödinger equation reads: i! dψ dt =! d Ψ m dx + VΨ + V 0Ψ.
More informationLandau-Zener Transition
Landau-Zener Transition David Chen June 1, 013 We want to solve the Landau-Zener problem [1], which consists in solving the two-level problem when the frequency of the external field ω(t varies across
More informationNilpotentni operatori i matrice
Sveučilište J. J. Strossmayera u Osijeku Odjel za matematiku Sveučilišni preddiplomski studij matematike Nikolina Romić Nilpotentni operatori i matrice Završni rad Osijek, 2016. Sveučilište J. J. Strossmayera
More informationDEVELOPMENT OF MATHEMATICAL MODELS TO PREDICT THE EFFECT OF INPUT PARAMETERS ON FEED RATE OF A RECIPROCATORY TUBE FUNNEL FEEDER
http://doi.org/10.24867/jpe-2018-01-067 JPE (2018) Vol.21 (1) Jain, A., Bansal, P., Khanna, P. Preliminary Note DEVELOPMENT OF MATHEMATICAL MODELS TO PREDICT THE EFFECT OF INPUT PARAMETERS ON FEED RATE
More informationFrom Quantum to Matter 2005
From Quantum to Matter 2005 Ronald Griessen Vrije Universiteit, Amsterdam AMOLF, May 24, 2004 vrije Universiteit amsterdam Why such a course? From Quantum to Matter: The main themes Wave functions Molecules
More informationLecture 4.6: Some special orthogonal functions
Lecture 4.6: Some special orthogonal functions Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 4340, Advanced Engineering Mathematics
More informationATOMSKA APSORP SORPCIJSKA TROSKOP
ATOMSKA APSORP SORPCIJSKA SPEKTROS TROSKOP OPIJA Written by Bette Kreuz Produced by Ruth Dusenbery University of Michigan-Dearborn 2000 Apsorpcija i emisija svjetlosti Fizika svjetlosti Spectroskopija
More informationProces Drella i Yana i potraga za te²kim esticama na hadronskim sudariva ima
Proces Drella i Yana i potraga za te²kim esticama na hadronskim sudariva ima Mentor: izv. prof. dr. sc. Kre²imir Kumeri ki Prirodoslovno-matemati ki fakultet, Fizi ki odsjek Sveu ili²te u Zagrebu velja
More informationPRIPADNOST RJEŠENJA KVADRATNE JEDNAČINE DANOM INTERVALU
MAT KOL Banja Luka) ISSN 0354 6969 p) ISSN 1986 58 o) Vol. XXI )015) 105 115 http://www.imvibl.org/dmbl/dmbl.htm PRIPADNOST RJEŠENJA KVADRATNE JEDNAČINE DANOM INTERVALU Bernadin Ibrahimpašić 1 Senka Ibrahimpašić
More informationSlika 1. Slika 2. Da ne bismo stalno izbacivali elemente iz skupa, mi ćemo napraviti još jedan niz markirano, gde će
Permutacije Zadatak. U vreći se nalazi n loptica različitih boja. Iz vreće izvlačimo redom jednu po jednu lopticu i stavljamo jednu pored druge. Koliko različitih redosleda boja možemo da dobijemo? Primer
More informationDiplomski rad br. 1396
Sveučilište u Zagrebu Fakultet elektrotehnike i računarstva Zavod za elektroniku, mikroelektroniku, računalne i inteligentne sustave Diplomski rad br. 1396 Uporaba višeslonog perceptrona za raspoznavane
More informationThe Bond Number Relationship for the O-H... O Systems
CROATICA CHEMICA ACTA CCACAA 61 (4) 815-819 (1988) CCA-1828 YU ISSN 0011-1643 UDC 541.571.9 Original Scientific Paper The Bond Number Relationship for the O-H... O Systems Slawomir J. Grabowski Institute
More informationJasna Kellner. snake. bee. mole. owl. branch. birds. bear. leaves. forest den. tree. sun. badger. butterfly
From the list of words below, fill in the blank boxes below each picture. 1 jazavac 5 lišće 9 proljeće 13 stablo 17 šuma 2 krtica 6 medvjed 10 ptice 14 sunce 18 med 3 lastavica 7 pećina 11 snijeg 15 životinje
More informationQuasi-Newtonove metode
Sveučilište J. J. Strossmayera u Osijeku Odjel za matematiku Milan Milinčević Quasi-Newtonove metode Završni rad Osijek, 2016. Sveučilište J. J. Strossmayera u Osijeku Odjel za matematiku Milan Milinčević
More informationPh2b Quiz - 2. Instructions
Ph2b Quiz - 2 Instructions 1. Your solutions are due by Monday, February 26th, 2018 at 4pm in the quiz box outside 201 E. Bridge. 2. Late quizzes will not be accepted, except in very special circumstances.
More informationGriffiths Chapter 1. Dan Wysocki. February 12, = A exp( λu 2 ) d u. e cx2 d x = 1 = λ = A = π. λ exp( λ(x a)2 ) λ exp( λ(x a)2 ) = πa2.
Griffiths Chapter 1 Dan Wysocki February 12, 215 Problem 1. Consider the gaussian distribution ρ) = A ep λ a) 2 ), where A, a, and λ are positive real constants. a. Use Equation 1.16 to determine A. 1
More informationTHE JONES-MUELLER TRANSFORMATION CARLOS HUNTE
Printed ISSN 33 8 Online ISSN 333 95 CD ISSN 333 839 CODEN FIZAE4 THE JONES-MUELLER TRANSFORMATION CARLOS HUNTE University of The West Indies, Cave Hill Campus, The Department of Computer Science, Mathematics
More informationChapter 5.3: Series solution near an ordinary point
Chapter 5.3: Series solution near an ordinary point We continue to study ODE s with polynomial coefficients of the form: P (x)y + Q(x)y + R(x)y = 0. Recall that x 0 is an ordinary point if P (x 0 ) 0.
More informationSTRESS OF ANGLE SECTION SUBJECTED TO TRANSVERSAL LOADING ACTING OUT OF THE SHEAR CENTER
STRESS OF ANGLE SECTION SUBJECTED TO TRANSVERSAL LOADING ACTING OUT OF THE SHEAR CENTER Filip Anić Josip Juraj Strossmayer University of Osijek, Faculty of Civil Engineering Osijek, Student Davorin Penava
More informationOn Acyclic Polynomials of [N]-Heteroannulenes
CROATICA CHEMICA ACTA CCACAA 54 (1) 91-95 (1981) CCA-1262 On Acyclic Polynomials of [N]-Heteroannulenes A. Graovac, D. Kasum, and N. Trinajstic YU ISSN 0011-1643 UDC 539.19 Note The Rugjer Boskovic Institute,
More informationSecond Year Electromagnetism Summer 2018 Caroline Terquem. Vacation work: Problem set 0. Revisions
Second Year Electromagnetism Summer 2018 Caroline Terquem Vacation work: Problem set 0 Revisions At the start of the second year, you will receive the second part of the Electromagnetism course. This vacation
More informationNIZOVI I REDOVI FUNKCIJA
SVEUČILIŠTE U ZAGREBU PRIRODOSLOVNO MATEMATIČKI FAKULTET MATEMATIČKI ODSJEK Danijela Piškor NIZOVI I REDOVI FUNKCIJA Diplomski rad Voditelj rada: izv. prof. dr. sc. Ljiljana Arambašić Zagreb, rujan 206.
More informationCategories and Filtrations
Categories and Filtrations Ludmil Katzarkov University of Miami July 10, 2017 1 / 1 Overview Navier-Stokes problems and Denaturation of DNA Burnside problem Uniformization 2 / 1 Navier-Stokes For V - velocity,
More informationProblems and Multiple Choice Questions
Problems and Multiple Choice Questions 1. A momentum operator in one dimension is 2. A position operator in 3 dimensions is 3. A kinetic energy operator in 1 dimension is 4. If two operator commute, a)
More informationInternational Journal of Scientific and Engineering Research, Volume 7, Issue 9,September-2016 ISSN
1574 International Journal of Scientific and Engineering Research, Volume 7, Issue 9,September-216 Energy and Wave-function correction for a quantum system after a small perturbation Samuel Mulugeta Bantikum
More informationConditional stability of Larkin methods with non-uniform grids
Theoret. Appl. Mech., Vol.37, No., pp.139-159, Belgrade 010 Conditional stability of Larkin methods with non-uniform grids Kazuhiro Fukuyo Abstract Stability analysis based on the von Neumann method showed
More informationFunkcijske jednadºbe
MEMO pripreme 2015. Marin Petkovi, 9. 6. 2015. Funkcijske jednadºbe Uvod i osnovne ideje U ovom predavanju obradit emo neke poznate funkcijske jednadºbe i osnovne ideje rje²avanja takvih jednadºbi. Uobi
More informationPARABOLIC POTENTIAL WELL
APPENDIX E PARABOLIC POTENTIAL WELL An example of an extremely important class of one-dimensional bound state in quantum mechanics is the simple harmonic oscillator whose potential can be written as V(x)=
More informationPrsten cijelih brojeva
SVEUČILIŠTE JOSIPA JURJA STROSSMAYERA U OSIJEKU ODJEL ZA MATEMATIKU Marijana Pravdić Prsten cijelih brojeva Diplomski rad Osijek, 2017. SVEUČILIŠTE JOSIPA JURJA STROSSMAYERA U OSIJEKU ODJEL ZA MATEMATIKU
More informationAn Exactly Solvable 3 Body Problem
An Exactly Solvable 3 Body Problem The most famous n-body problem is one where particles interact by an inverse square-law force. However, there is a class of exactly solvable n-body problems in which
More informationHornerov algoritam i primjene
Osječki matematički list 7(2007), 99 106 99 STUDENTSKA RUBRIKA Hornerov algoritam i primjene Zoran Tomljanović Sažetak. U ovom članku obrad uje se Hornerov algoritam za efikasno računanje vrijednosti polinoma
More informationEXPERIMENTAL ANALYSIS OF THE STRENGTH OF A POLYMER PRODUCED FROM RECYCLED MATERIAL
A. Jurić et al. EXPERIMENTAL ANALYSIS OF THE STRENGTH OF A POLYMER PRODUCED FROM RECYCLED MATERIAL Aleksandar Jurić, Tihomir Štefić, Zlatko Arbanas ISSN 10-651 UDC/UDK 60.17.1/.:678.74..017 Preliminary
More informationHolomorphic functions which preserve holomorphic semigroups
Holomorphic functions which preserve holomorphic semigroups University of Oxford London Mathematical Society Regional Meeting Birmingham, 15 September 2016 Heat equation u t = xu (x Ω R d, t 0), u(t, x)
More informationALGEBRAIC MODELLING OF QUANTUM MECHANICAL EQUATIONS IN THE FINITE- AND INFINITE-DIMENSIONAL HILBERT SPACES
PHYSICS DEPARTMENT OF THE FACULTY OF SCIENCE Norman Dwight Megill ALGEBRAIC MODELLING OF QUANTUM MECHANICAL EQUATIONS IN THE FINITE- AND INFINITE-DIMENSIONAL HILBERT SPACES DOCTORAL THESIS Zagreb, 2011
More informationPhysics 137A Quantum Mechanics Fall 2012 Midterm II - Solutions
Physics 37A Quantum Mechanics Fall 0 Midterm II - Solutions These are the solutions to the exam given to Lecture Problem [5 points] Consider a particle with mass m charge q in a simple harmonic oscillator
More informationTWO-STEP LORENTZ TRANSFORMATION OF FORCE. AMON ILAKOVAC and LUKA POPOV
Printed ISSN 1330 0008 Online ISSN 1333 9125 CD ISSN 1333 8390 CODEN FIZAE4 TWO-STEP LORENTZ TRANSFORMATION OF FORCE AMON ILAKOVAC and LUKA POPOV Department of Physics, Faculty of Science, University of
More informationNucleus. Electron Cloud
Atomic Structure I. Picture of an Atom Nucleus Electron Cloud II. Subatomic particles Particle Symbol Charge Relative Mass (amu) protons p + +1 1.0073 neutrons n 0 1.0087 electrons e - -1 0.00054858 Compare
More informationElectric and Magnetic Forces in Lagrangian and Hamiltonian Formalism
Electric and Magnetic Forces in Lagrangian and Hamiltonian Formalism Benjamin Hornberger 1/26/1 Phy 55, Classical Electrodynamics, Prof. Goldhaber Lecture notes from Oct. 26, 21 Lecture held by Prof. Weisberger
More informationChem 6 Sample exam 2 (150 points total) NAME:
hem 6 Sample exam 2 (150 points total) @ This is a closed book exam to which the onor Principle applies. @ The last page contains equations and physical constants; you can detach it for easy reference.
More informationHistory of the Periodic Table
SECTION 5.1 History of the Periodic Table By 60, more than 60 elements had been discovered. Chemists had to learn the properties of these elements as well as those of the many compounds that the elements
More informationCyclical Surfaces Created by a Conical Helix
Professional paper Accepted 23.11.2007. TATIANA OLEJNÍKOVÁ Cyclical Surfaces Created by a Conical Helix Cyclical Surfaces Created by a Conical Helix ABSTRACT The paper describes cyclical surfaces created
More informationProgramiranje u realnom vremenu Bojan Furlan
Programiranje u realnom vremenu Bojan Furlan Tri procesa sa D = T imaju sledeće karakteristike: Proces T C a 3 1 b 6 2 c 18 5 (a) Pokazati kako se može konstruisati ciklično izvršavanje ovih procesa. (b)
More informationPrimjena numeričke metode Runge-Kutta na rješavanje problema početnih i rubnih uvjeta
SVEUČILIŠTE U ZAGREBU PRIRODOSLOVNO-MATEMATIČKI FAKULTET FIZIČKI ODSJEK SMJER: PROFESOR FIZIKE I INFORMATIKE Ivan Banić Diplomski rad Primjena numeričke metode Runge-Kutta na rješavanje problema početnih
More informationThe temperature dependence of the disproportionation reaction of iodous acid in aqueous sulfuric acid solutions
J. Serb. Chem. Soc. 67(5)347 351(2002) UDC 542.9:546.155+535.243:536.5 JSCS-2955 Original scientific paper The temperature dependence of the disproportionation reaction of iodous acid in aqueous sulfuric
More informationRate Equation Model for Semiconductor Lasers
Rate Equation Model for Semiconductor Lasers Prof. Sebastian Wieczorek, Applied Mathematics, University College Cork October 21, 2015 1 Introduction Let s consider a laser which consist of an optical resonator
More information